
where \(\mathbf {X}\in \mathcal {H}_1\)  and \(\mathbf {V}\in \mathcal {H}_2\)  are the primal variables, \(\mathbf {U}\in \mathcal {H}_2\)  is the dual variable and the norm used in the updates is induced by some arbitrary inner product \(\left\langle \cdot ,\cdot \right\rangle _\ast \)  defined on \(\mathcal {H}_2\) .
By adapting the following convergence result in [1]}, one can show that the above ADMM algorithm solves (REF ).
